harmonics: I 0 (A 0 ), I 1 (A 0 ), I 2 (A 0 ), where coefficients depend on pumping and are
defined with the help of the Fourier transform, for instance, as it is shown in example
of WAC approximation by the quadratic function.
The effective operation is performed with zero or small cutoff angles and at large
excesses of the pumping current above the threshold values, since for the laser
diodes, the mode with negative pumping currents leads to the QWLD failure. The
modulation index defining as the ratio of the amplitude of AC component of the
fundamental harmonic to the DC component is from 0.1 to 0.2.
The constant phase incursions ϕ 0O , ϕ 10O , ϕ 20O of harmonics play the special role
in analysis of the OEO DM system. Their values on the PD area are determined by
the phase-frequency characteristic shape (we mean the PFC for optical harmonics) of
the laser and the PFC of the optical filter applied for selection. The fundamental
harmonic level (on the radio frequency f ) in the PD photocurrent I 1 (A 0 ) is proportional to cos(ϕ 10O À ϕ 0O ). At that, for optical phases ϕ 0O , ϕ 10O , ϕ 20O and their
differences, the following relation is true: ϕ 0O ¼ 2πν 0 T OF20 , ϕ 10O ¼ 2π(ν 0 + f )T OF10 ,
Δϕ 0e ¼ ϕ 10O À ϕ 0O ¼ 2πν 0 T OF20 À 2π(ν 0 + f )T OF10 . Here we specially introduce
the fictive constant delays T OF10 and T OF20 in order to be more clear to understand
how self-heterodyning occurs in OEO with QWLD DM.
The mathematical expression E LL for the sum of EMF two harmonics on the PD
area after selection by the optical filter can be written as:
E LL ¼ E 0L Á k 0 Á I 0 A 0
ð Þ Á cos 2πtν 0 þ ϕ 0e þ φ em t
ð Þ
ð
Þ
þE 0L Á k 1 Á I 1 A 0
ð Þ Á cos 2πt ν 0 À f
ð
ÞÀϕ 0e À φ em t
ð Þ
ð
Þ
,
ð3:22Þ
Now we find the ratio: K p ¼ E PD /[2E 0L Á k 0 Á I 0 (A 0 )]. Assuming that amplitudes of
harmonics ν 0 and ν 0 À f after passing through the optical filter are equal:
E 0L Á k 0 Á I 0 (A 0 ) ¼ E 0L Á k 1 Á I 1 (A 0 ), and using the trigonometric formula for adding
the cosines of different angles, we obtain:
K P ¼ cos 2πtν 0 þ ϕ 0e þ φ em t
ð Þ
½
þ cos 2πt ν 0 À f
ð
ÞÀϕ 0e À φ em t
ð Þ
½
f
g
¼ 2 cos
2πtν 0 þ ϕ 0e þ φ em t
ð Þ þ 2πt ν 0 À f
ð
ÞÀϕ 0e À φ em t
ð Þ
2
!
 cos
2πtν 0 þ ϕ 0e þ φ em t
ð Þ À 2πt ν 0 À f
ð
ÞÀϕ 0e À φ em t
ð Þ
½
2
!
:
ð3:23Þ
Transforming the last expression for K p , we have:
K P ¼ 2 cos
2 Á 2πtν 0 À 2πtf þ Δϕ 0e þ φ em t
ð Þ À φ em t
ð Þ
2
!
Á cos 2πtf þ Δϕ 0e þ φ em t
ð Þ
½
:
ð3:24Þ
The first multiplier in Eq. (3.23) is the oscillating term in time with the average
value equaled to 1.
100 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .
defined with the help of the Fourier transform, for instance, as it is shown in example
of WAC approximation by the quadratic function.
The effective operation is performed with zero or small cutoff angles and at large
excesses of the pumping current above the threshold values, since for the laser
diodes, the mode with negative pumping currents leads to the QWLD failure. The
modulation index defining as the ratio of the amplitude of AC component of the
fundamental harmonic to the DC component is from 0.1 to 0.2.
The constant phase incursions ϕ 0O , ϕ 10O , ϕ 20O of harmonics play the special role
in analysis of the OEO DM system. Their values on the PD area are determined by
the phase-frequency characteristic shape (we mean the PFC for optical harmonics) of
the laser and the PFC of the optical filter applied for selection. The fundamental
harmonic level (on the radio frequency f ) in the PD photocurrent I 1 (A 0 ) is proportional to cos(ϕ 10O À ϕ 0O ). At that, for optical phases ϕ 0O , ϕ 10O , ϕ 20O and their
differences, the following relation is true: ϕ 0O ¼ 2πν 0 T OF20 , ϕ 10O ¼ 2π(ν 0 + f )T OF10 ,
Δϕ 0e ¼ ϕ 10O À ϕ 0O ¼ 2πν 0 T OF20 À 2π(ν 0 + f )T OF10 . Here we specially introduce
the fictive constant delays T OF10 and T OF20 in order to be more clear to understand
how self-heterodyning occurs in OEO with QWLD DM.
The mathematical expression E LL for the sum of EMF two harmonics on the PD
area after selection by the optical filter can be written as:
E LL ¼ E 0L Á k 0 Á I 0 A 0
ð Þ Á cos 2πtν 0 þ ϕ 0e þ φ em t
ð Þ
ð
Þ
þE 0L Á k 1 Á I 1 A 0
ð Þ Á cos 2πt ν 0 À f
ð
ÞÀϕ 0e À φ em t
ð Þ
ð
Þ
,
ð3:22Þ
Now we find the ratio: K p ¼ E PD /[2E 0L Á k 0 Á I 0 (A 0 )]. Assuming that amplitudes of
harmonics ν 0 and ν 0 À f after passing through the optical filter are equal:
E 0L Á k 0 Á I 0 (A 0 ) ¼ E 0L Á k 1 Á I 1 (A 0 ), and using the trigonometric formula for adding
the cosines of different angles, we obtain:
K P ¼ cos 2πtν 0 þ ϕ 0e þ φ em t
ð Þ
½
þ cos 2πt ν 0 À f
ð
ÞÀϕ 0e À φ em t
ð Þ
½
f
g
¼ 2 cos
2πtν 0 þ ϕ 0e þ φ em t
ð Þ þ 2πt ν 0 À f
ð
ÞÀϕ 0e À φ em t
ð Þ
2
!
 cos
2πtν 0 þ ϕ 0e þ φ em t
ð Þ À 2πt ν 0 À f
ð
ÞÀϕ 0e À φ em t
ð Þ
½
2
!
:
ð3:23Þ
Transforming the last expression for K p , we have:
K P ¼ 2 cos
2 Á 2πtν 0 À 2πtf þ Δϕ 0e þ φ em t
ð Þ À φ em t
ð Þ
2
!
Á cos 2πtf þ Δϕ 0e þ φ em t
ð Þ
½
:
ð3:24Þ
The first multiplier in Eq. (3.23) is the oscillating term in time with the average
value equaled to 1.
100 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .